LMFDB - Embedded newform 672.4.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(1,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{43}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 43 $ x^2 - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.55744$ of defining polynomial
Character	$\chi$	$=$	672.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +21.1149 q^{5} +7.00000 q^{7} +9.00000 q^{9} -11.1149 q^{11} +26.0000 q^{13} -63.3446 q^{15} +67.3446 q^{17} +42.2298 q^{19} -21.0000 q^{21} +122.264 q^{23} +320.838 q^{25} -27.0000 q^{27} -16.2298 q^{29} -279.149 q^{31} +33.3446 q^{33} +147.804 q^{35} -123.838 q^{37} -78.0000 q^{39} -32.2636 q^{41} -281.838 q^{43} +190.034 q^{45} -250.689 q^{47} +49.0000 q^{49} -202.034 q^{51} -27.8380 q^{53} -234.689 q^{55} -126.689 q^{57} +677.906 q^{59} +73.9322 q^{61} +63.0000 q^{63} +548.987 q^{65} +845.446 q^{67} -366.791 q^{69} +739.250 q^{71} -702.203 q^{73} -962.514 q^{75} -77.8041 q^{77} +250.689 q^{79} +81.0000 q^{81} -699.352 q^{83} +1421.97 q^{85} +48.6893 q^{87} +489.574 q^{89} +182.000 q^{91} +837.446 q^{93} +891.676 q^{95} +1184.37 q^{97} -100.034 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 16 q^{5} + 14 q^{7} + 18 q^{9} + 4 q^{11} + 52 q^{13} - 48 q^{15} + 56 q^{17} + 32 q^{19} - 42 q^{21} - 44 q^{23} + 222 q^{25} - 54 q^{27} + 20 q^{29} - 296 q^{31} - 12 q^{33} + 112 q^{35}+ \cdots + 36 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 16 * q^5 + 14 * q^7 + 18 * q^9 + 4 * q^11 + 52 * q^13 - 48 * q^15 + 56 * q^17 + 32 * q^19 - 42 * q^21 - 44 * q^23 + 222 * q^25 - 54 * q^27 + 20 * q^29 - 296 * q^31 - 12 * q^33 + 112 * q^35 + 172 * q^37 - 156 * q^39 + 224 * q^41 - 144 * q^43 + 144 * q^45 - 344 * q^47 + 98 * q^49 - 168 * q^51 + 364 * q^53 - 312 * q^55 - 96 * q^57 + 464 * q^59 + 620 * q^61 + 126 * q^63 + 416 * q^65 + 904 * q^67 + 132 * q^69 + 508 * q^71 + 12 * q^73 - 666 * q^75 + 28 * q^77 + 344 * q^79 + 162 * q^81 + 280 * q^83 + 1480 * q^85 - 60 * q^87 + 848 * q^89 + 364 * q^91 + 888 * q^93 + 944 * q^95 + 1372 * q^97 + 36 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	21.1149	1.88857	0.944286	−	0.329126i	$-0.106754\pi$
$5$	21.1149	1.88857	0.944286	+	0.329126i	$0.106754\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.1149	−0.304660	−0.152330	−	0.988330i	$-0.548678\pi$
$11$	−11.1149	−0.304660	−0.152330	+	0.988330i	$0.548678\pi$
$12$	0	0
$13$	26.0000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	26.0000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−63.3446	−1.09037
$16$	0	0
$17$	67.3446	0.960792	0.480396	−	0.877052i	$-0.340493\pi$
$17$	67.3446	0.960792	0.480396	+	0.877052i	$0.340493\pi$
$18$	0	0
$19$	42.2298	0.509904	0.254952	−	0.966954i	$-0.417940\pi$
$19$	42.2298	0.509904	0.254952	+	0.966954i	$0.417940\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	122.264	1.10842	0.554212	−	0.832376i	$-0.313020\pi$
$23$	122.264	1.10842	0.554212	+	0.832376i	$0.313020\pi$
$24$	0	0
$25$	320.838	2.56670
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−16.2298	−0.103924	−0.0519619	−	0.998649i	$-0.516547\pi$
$29$	−16.2298	−0.103924	−0.0519619	+	0.998649i	$0.516547\pi$
$30$	0	0
$31$	−279.149	−1.61731	−0.808655	−	0.588283i	$-0.799804\pi$
$31$	−279.149	−1.61731	−0.808655	+	0.588283i	$0.799804\pi$
$32$	0	0
$33$	33.3446	0.175896
$34$	0	0
$35$	147.804	0.713813
$36$	0	0
$37$	−123.838	−0.550239	−0.275120	−	0.961410i	$-0.588717\pi$
$37$	−123.838	−0.550239	−0.275120	+	0.961410i	$0.588717\pi$
$38$	0	0
$39$	−78.0000	−0.320256
$40$	0	0
$41$	−32.2636	−0.122896	−0.0614480	−	0.998110i	$-0.519572\pi$
$41$	−32.2636	−0.122896	−0.0614480	+	0.998110i	$0.519572\pi$
$42$	0	0
$43$	−281.838	−0.999532	−0.499766	−	0.866160i	$-0.666581\pi$
$43$	−281.838	−0.999532	−0.499766	+	0.866160i	$0.666581\pi$
$44$	0	0
$45$	190.034	0.629524
$46$	0	0
$47$	−250.689	−0.778017	−0.389008	−	0.921234i	$-0.627182\pi$
$47$	−250.689	−0.778017	−0.389008	+	0.921234i	$0.627182\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−202.034	−0.554714
$52$	0	0
$53$	−27.8380	−0.0721481	−0.0360740	−	0.999349i	$-0.511485\pi$
$53$	−27.8380	−0.0721481	−0.0360740	+	0.999349i	$0.511485\pi$
$54$	0	0
$55$	−234.689	−0.575373
$56$	0	0
$57$	−126.689	−0.294393
$58$	0	0
$59$	677.906	1.49586	0.747931	−	0.663777i	$-0.231048\pi$
$59$	677.906	1.49586	0.747931	+	0.663777i	$0.231048\pi$
$60$	0	0
$61$	73.9322	0.155181	0.0775906	−	0.996985i	$-0.475277\pi$
$61$	73.9322	0.155181	0.0775906	+	0.996985i	$0.475277\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	548.987	1.04759
$66$	0	0
$67$	845.446	1.54161	0.770804	−	0.637073i	$-0.219855\pi$
$67$	845.446	1.54161	0.770804	+	0.637073i	$0.219855\pi$
$68$	0	0
$69$	−366.791	−0.639948
$70$	0	0
$71$	739.250	1.23567	0.617837	−	0.786306i	$-0.288009\pi$
$71$	739.250	1.23567	0.617837	+	0.786306i	$0.288009\pi$
$72$	0	0
$73$	−702.203	−1.12585	−0.562923	−	0.826510i	$-0.690323\pi$
$73$	−702.203	−1.12585	−0.562923	+	0.826510i	$0.690323\pi$
$74$	0	0
$75$	−962.514	−1.48189
$76$	0	0
$77$	−77.8041	−0.115151
$78$	0	0
$79$	250.689	0.357022	0.178511	−	0.983938i	$-0.442872\pi$
$79$	250.689	0.357022	0.178511	+	0.983938i	$0.442872\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−699.352	−0.924866	−0.462433	−	0.886654i	$-0.653023\pi$
$83$	−699.352	−0.924866	−0.462433	+	0.886654i	$0.653023\pi$
$84$	0	0
$85$	1421.97	1.81453
$86$	0	0
$87$	48.6893	0.0600004
$88$	0	0
$89$	489.574	0.583087	0.291544	−	0.956558i	$-0.405831\pi$
$89$	489.574	0.583087	0.291544	+	0.956558i	$0.405831\pi$
$90$	0	0
$91$	182.000	0.209657
$92$	0	0
$93$	837.446	0.933754
$94$	0	0
$95$	891.676	0.962990
$96$	0	0
$97$	1184.37	1.23973	0.619866	−	0.784707i	$-0.287187\pi$
$97$	1184.37	1.23973	0.619866	+	0.784707i	$0.287187\pi$
$98$	0	0
$99$	−100.034	−0.101553
$100$	0	0
$101$	1554.56	1.53153	0.765765	−	0.643120i	$-0.222360\pi$
$101$	1554.56	1.53153	0.765765	+	0.643120i	$0.222360\pi$
$102$	0	0
$103$	−1780.01	−1.70282	−0.851408	−	0.524504i	$-0.824251\pi$
$103$	−1780.01	−1.70282	−0.851408	+	0.524504i	$0.824251\pi$
$104$	0	0
$105$	−443.412	−0.412120
$106$	0	0
$107$	1140.37	1.03032	0.515159	−	0.857095i	$-0.327733\pi$
$107$	1140.37	1.03032	0.515159	+	0.857095i	$0.327733\pi$
$108$	0	0
$109$	2089.68	1.83628	0.918141	−	0.396255i	$-0.129690\pi$
$109$	2089.68	1.83628	0.918141	+	0.396255i	$0.129690\pi$
$110$	0	0
$111$	371.514	0.317681
$112$	0	0
$113$	−332.298	−0.276636	−0.138318	−	0.990388i	$-0.544170\pi$
$113$	−332.298	−0.276636	−0.138318	+	0.990388i	$0.544170\pi$
$114$	0	0
$115$	2581.58	2.09334
$116$	0	0
$117$	234.000	0.184900
$118$	0	0
$119$	471.412	0.363145
$120$	0	0
$121$	−1207.46	−0.907182
$122$	0	0
$123$	96.7909	0.0709540
$124$	0	0
$125$	4135.10	2.95883
$126$	0	0
$127$	−1261.72	−0.881569	−0.440785	−	0.897613i	$-0.645300\pi$
$127$	−1261.72	−0.881569	−0.440785	+	0.897613i	$0.645300\pi$
$128$	0	0
$129$	845.514	0.577080
$130$	0	0
$131$	−2639.03	−1.76010	−0.880049	−	0.474882i	$-0.842491\pi$
$131$	−2639.03	−1.76010	−0.880049	+	0.474882i	$0.842491\pi$
$132$	0	0
$133$	295.608	0.192725
$134$	0	0
$135$	−570.102	−0.363456
$136$	0	0
$137$	−2000.18	−1.24735	−0.623674	−	0.781685i	$-0.714361\pi$
$137$	−2000.18	−1.24735	−0.623674	+	0.781685i	$0.714361\pi$
$138$	0	0
$139$	2014.89	1.22950	0.614752	−	0.788721i	$-0.289256\pi$
$139$	2014.89	1.22950	0.614752	+	0.788721i	$0.289256\pi$
$140$	0	0
$141$	752.068	0.449188
$142$	0	0
$143$	−288.987	−0.168995
$144$	0	0
$145$	−342.689	−0.196268
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	140.810	0.0774201	0.0387100	−	0.999250i	$-0.487675\pi$
$149$	140.810	0.0774201	0.0387100	+	0.999250i	$0.487675\pi$
$150$	0	0
$151$	2493.00	1.34356	0.671780	−	0.740751i	$-0.265530\pi$
$151$	2493.00	1.34356	0.671780	+	0.740751i	$0.265530\pi$
$152$	0	0
$153$	606.102	0.320264
$154$	0	0
$155$	−5894.19	−3.05441
$156$	0	0
$157$	2681.28	1.36299	0.681496	−	0.731822i	$-0.261330\pi$
$157$	2681.28	1.36299	0.681496	+	0.731822i	$0.261330\pi$
$158$	0	0
$159$	83.5141	0.0416547
$160$	0	0
$161$	855.846	0.418945
$162$	0	0
$163$	3625.26	1.74204	0.871019	−	0.491250i	$-0.163460\pi$
$163$	3625.26	1.74204	0.871019	+	0.491250i	$0.163460\pi$
$164$	0	0
$165$	704.068	0.332192
$166$	0	0
$167$	−1242.82	−0.575884	−0.287942	−	0.957648i	$-0.592971\pi$
$167$	−1242.82	−0.575884	−0.287942	+	0.957648i	$0.592971\pi$
$168$	0	0
$169$	−1521.00	−0.692308
$170$	0	0
$171$	380.068	0.169968
$172$	0	0
$173$	249.778	0.109770	0.0548851	−	0.998493i	$-0.482521\pi$
$173$	249.778	0.109770	0.0548851	+	0.998493i	$0.482521\pi$
$174$	0	0
$175$	2245.87	0.970123
$176$	0	0
$177$	−2033.72	−0.863636
$178$	0	0
$179$	−3529.48	−1.47378	−0.736888	−	0.676015i	$-0.763705\pi$
$179$	−3529.48	−1.47378	−0.736888	+	0.676015i	$0.763705\pi$
$180$	0	0
$181$	−2359.73	−0.969047	−0.484524	−	0.874778i	$-0.661007\pi$
$181$	−2359.73	−0.969047	−0.484524	+	0.874778i	$0.661007\pi$
$182$	0	0
$183$	−221.797	−0.0895939
$184$	0	0
$185$	−2614.82	−1.03917
$186$	0	0
$187$	−748.527	−0.292715
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−686.859	−0.260206	−0.130103	−	0.991500i	$-0.541531\pi$
$191$	−686.859	−0.260206	−0.130103	+	0.991500i	$0.541531\pi$
$192$	0	0
$193$	−1456.89	−0.543365	−0.271682	−	0.962387i	$-0.587580\pi$
$193$	−1456.89	−0.543365	−0.271682	+	0.962387i	$0.587580\pi$
$194$	0	0
$195$	−1646.96	−0.604827
$196$	0	0
$197$	−4741.71	−1.71489	−0.857443	−	0.514579i	$-0.827948\pi$
$197$	−4741.71	−1.71489	−0.857443	+	0.514579i	$0.827948\pi$
$198$	0	0
$199$	563.269	0.200649	0.100324	−	0.994955i	$-0.468012\pi$
$199$	563.269	0.200649	0.100324	+	0.994955i	$0.468012\pi$
$200$	0	0
$201$	−2536.34	−0.890048
$202$	0	0
$203$	−113.608	−0.0392795
$204$	0	0
$205$	−681.243	−0.232098
$206$	0	0
$207$	1100.37	0.369474
$208$	0	0
$209$	−469.379	−0.155347
$210$	0	0
$211$	3241.19	1.05750	0.528751	−	0.848777i	$-0.322661\pi$
$211$	3241.19	1.05750	0.528751	+	0.848777i	$0.322661\pi$
$212$	0	0
$213$	−2217.75	−0.713417
$214$	0	0
$215$	−5950.98	−1.88769
$216$	0	0
$217$	−1954.04	−0.611286
$218$	0	0
$219$	2106.61	0.650007
$220$	0	0
$221$	1750.96	0.532952
$222$	0	0
$223$	2656.65	0.797768	0.398884	−	0.917001i	$-0.369398\pi$
$223$	2656.65	0.797768	0.398884	+	0.917001i	$0.369398\pi$
$224$	0	0
$225$	2887.54	0.855568
$226$	0	0
$227$	3125.02	0.913721	0.456860	−	0.889538i	$-0.348974\pi$
$227$	3125.02	0.913721	0.456860	+	0.889538i	$0.348974\pi$
$228$	0	0
$229$	−403.107	−0.116324	−0.0581618	−	0.998307i	$-0.518524\pi$
$229$	−403.107	−0.116324	−0.0581618	+	0.998307i	$0.518524\pi$
$230$	0	0
$231$	233.412	0.0664823
$232$	0	0
$233$	−1156.47	−0.325163	−0.162581	−	0.986695i	$-0.551982\pi$
$233$	−1156.47	−0.325163	−0.162581	+	0.986695i	$0.551982\pi$
$234$	0	0
$235$	−5293.27	−1.46934
$236$	0	0
$237$	−752.068	−0.206127
$238$	0	0
$239$	3733.02	1.01033	0.505165	−	0.863023i	$-0.331432\pi$
$239$	3733.02	1.01033	0.505165	+	0.863023i	$0.331432\pi$
$240$	0	0
$241$	3153.04	0.842761	0.421380	−	0.906884i	$-0.361546\pi$
$241$	3153.04	0.842761	0.421380	+	0.906884i	$0.361546\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1034.63	0.269796
$246$	0	0
$247$	1097.97	0.282844
$248$	0	0
$249$	2098.06	0.533972
$250$	0	0
$251$	−2441.45	−0.613955	−0.306978	−	0.951717i	$-0.599318\pi$
$251$	−2441.45	−0.613955	−0.306978	+	0.951717i	$0.599318\pi$
$252$	0	0
$253$	−1358.95	−0.337692
$254$	0	0
$255$	−4265.92	−1.04762
$256$	0	0
$257$	2416.55	0.586538	0.293269	−	0.956030i	$-0.405257\pi$
$257$	2416.55	0.586538	0.293269	+	0.956030i	$0.405257\pi$
$258$	0	0
$259$	−866.866	−0.207971
$260$	0	0
$261$	−146.068	−0.0346413
$262$	0	0
$263$	509.427	0.119440	0.0597198	−	0.998215i	$-0.480979\pi$
$263$	509.427	0.119440	0.0597198	+	0.998215i	$0.480979\pi$
$264$	0	0
$265$	−587.797	−0.136257
$266$	0	0
$267$	−1468.72	−0.336646
$268$	0	0
$269$	−8309.43	−1.88340	−0.941701	−	0.336452i	$-0.890773\pi$
$269$	−8309.43	−1.88340	−0.941701	+	0.336452i	$0.890773\pi$
$270$	0	0
$271$	−1499.88	−0.336204	−0.168102	−	0.985770i	$-0.553764\pi$
$271$	−1499.88	−0.336204	−0.168102	+	0.985770i	$0.553764\pi$
$272$	0	0
$273$	−546.000	−0.121046
$274$	0	0
$275$	−3566.08	−0.781972
$276$	0	0
$277$	1724.03	0.373959	0.186980	−	0.982364i	$-0.440130\pi$
$277$	1724.03	0.373959	0.186980	+	0.982364i	$0.440130\pi$
$278$	0	0
$279$	−2512.34	−0.539103
$280$	0	0
$281$	−8393.23	−1.78184	−0.890922	−	0.454156i	$-0.849941\pi$
$281$	−8393.23	−1.78184	−0.890922	+	0.454156i	$0.849941\pi$
$282$	0	0
$283$	8049.99	1.69089	0.845445	−	0.534062i	$-0.179335\pi$
$283$	8049.99	1.69089	0.845445	+	0.534062i	$0.179335\pi$
$284$	0	0
$285$	−2675.03	−0.555982
$286$	0	0
$287$	−225.846	−0.0464503
$288$	0	0
$289$	−377.701	−0.0768778
$290$	0	0
$291$	−3553.10	−0.715760
$292$	0	0
$293$	9886.67	1.97128	0.985641	−	0.168854i	$-0.0540067\pi$
$293$	9886.67	1.97128	0.985641	+	0.168854i	$0.0540067\pi$
$294$	0	0
$295$	14313.9	2.82504
$296$	0	0
$297$	300.102	0.0586319
$298$	0	0
$299$	3178.85	0.614843
$300$	0	0
$301$	−1972.87	−0.377788
$302$	0	0
$303$	−4663.68	−0.884230
$304$	0	0
$305$	1561.07	0.293071
$306$	0	0
$307$	1721.15	0.319970	0.159985	−	0.987119i	$-0.448855\pi$
$307$	1721.15	0.319970	0.159985	+	0.987119i	$0.448855\pi$
$308$	0	0
$309$	5340.04	0.983122
$310$	0	0
$311$	−4417.94	−0.805525	−0.402762	−	0.915305i	$-0.631950\pi$
$311$	−4417.94	−0.805525	−0.402762	+	0.915305i	$0.631950\pi$
$312$	0	0
$313$	−8351.08	−1.50809	−0.754043	−	0.656825i	$-0.771899\pi$
$313$	−8351.08	−1.50809	−0.754043	+	0.656825i	$0.771899\pi$
$314$	0	0
$315$	1330.24	0.237938
$316$	0	0
$317$	5002.52	0.886339	0.443170	−	0.896438i	$-0.353854\pi$
$317$	5002.52	0.886339	0.443170	+	0.896438i	$0.353854\pi$
$318$	0	0
$319$	180.392	0.0316614
$320$	0	0
$321$	−3421.12	−0.594854
$322$	0	0
$323$	2843.95	0.489912
$324$	0	0
$325$	8341.79	1.42375
$326$	0	0
$327$	−6269.03	−1.06018
$328$	0	0
$329$	−1754.82	−0.294063
$330$	0	0
$331$	−1210.92	−0.201083	−0.100541	−	0.994933i	$-0.532057\pi$
$331$	−1210.92	−0.201083	−0.100541	+	0.994933i	$0.532057\pi$
$332$	0	0
$333$	−1114.54	−0.183413
$334$	0	0
$335$	17851.5	2.91144
$336$	0	0
$337$	611.974	0.0989209	0.0494604	−	0.998776i	$-0.484250\pi$
$337$	611.974	0.0989209	0.0494604	+	0.998776i	$0.484250\pi$
$338$	0	0
$339$	996.893	0.159716
$340$	0	0
$341$	3102.70	0.492730
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−7744.75	−1.20859
$346$	0	0
$347$	−8546.91	−1.32225	−0.661127	−	0.750274i	$-0.729922\pi$
$347$	−8546.91	−1.32225	−0.661127	+	0.750274i	$0.729922\pi$
$348$	0	0
$349$	−5144.82	−0.789101	−0.394550	−	0.918874i	$-0.629100\pi$
$349$	−5144.82	−0.789101	−0.394550	+	0.918874i	$0.629100\pi$
$350$	0	0
$351$	−702.000	−0.106752
$352$	0	0
$353$	−9998.93	−1.50762	−0.753809	−	0.657093i	$-0.771786\pi$
$353$	−9998.93	−1.50762	−0.753809	+	0.657093i	$0.771786\pi$
$354$	0	0
$355$	15609.2	2.33366
$356$	0	0
$357$	−1414.24	−0.209662
$358$	0	0
$359$	−3832.83	−0.563480	−0.281740	−	0.959491i	$-0.590911\pi$
$359$	−3832.83	−0.563480	−0.281740	+	0.959491i	$0.590911\pi$
$360$	0	0
$361$	−5075.65	−0.739998
$362$	0	0
$363$	3622.38	0.523762
$364$	0	0
$365$	−14826.9	−2.12624
$366$	0	0
$367$	−5782.49	−0.822462	−0.411231	−	0.911531i	$-0.634901\pi$
$367$	−5782.49	−0.822462	−0.411231	+	0.911531i	$0.634901\pi$
$368$	0	0
$369$	−290.373	−0.0409653
$370$	0	0
$371$	−194.866	−0.0272694
$372$	0	0
$373$	−9914.87	−1.37633	−0.688167	−	0.725552i	$-0.741584\pi$
$373$	−9914.87	−1.37633	−0.688167	+	0.725552i	$0.741584\pi$
$374$	0	0
$375$	−12405.3	−1.70828
$376$	0	0
$377$	−421.974	−0.0576465
$378$	0	0
$379$	−9678.39	−1.31173	−0.655865	−	0.754879i	$-0.727696\pi$
$379$	−9678.39	−1.31173	−0.655865	+	0.754879i	$0.727696\pi$
$380$	0	0
$381$	3785.15	0.508974
$382$	0	0
$383$	8619.74	1.14999	0.574997	−	0.818155i	$-0.305003\pi$
$383$	8619.74	1.14999	0.574997	+	0.818155i	$0.305003\pi$
$384$	0	0
$385$	−1642.82	−0.217470
$386$	0	0
$387$	−2536.54	−0.333177
$388$	0	0
$389$	−9220.61	−1.20181	−0.600904	−	0.799321i	$-0.705193\pi$
$389$	−9220.61	−1.20181	−0.600904	+	0.799321i	$0.705193\pi$
$390$	0	0
$391$	8233.80	1.06496
$392$	0	0
$393$	7917.08	1.01619
$394$	0	0
$395$	5293.27	0.674262
$396$	0	0
$397$	−12551.5	−1.58676	−0.793379	−	0.608728i	$-0.791680\pi$
$397$	−12551.5	−1.58676	−0.793379	+	0.608728i	$0.791680\pi$
$398$	0	0
$399$	−886.825	−0.111270
$400$	0	0
$401$	−11057.6	−1.37703	−0.688514	−	0.725223i	$-0.741737\pi$
$401$	−11057.6	−1.37703	−0.688514	+	0.725223i	$0.741737\pi$
$402$	0	0
$403$	−7257.87	−0.897122
$404$	0	0
$405$	1710.31	0.209841
$406$	0	0
$407$	1376.44	0.167636
$408$	0	0
$409$	−10535.3	−1.27368	−0.636842	−	0.770995i	$-0.719759\pi$
$409$	−10535.3	−1.27368	−0.636842	+	0.770995i	$0.719759\pi$
$410$	0	0
$411$	6000.53	0.720157
$412$	0	0
$413$	4745.34	0.565382
$414$	0	0
$415$	−14766.7	−1.74668
$416$	0	0
$417$	−6044.68	−0.709854
$418$	0	0
$419$	4846.31	0.565055	0.282527	−	0.959259i	$-0.408827\pi$
$419$	4846.31	0.565055	0.282527	+	0.959259i	$0.408827\pi$
$420$	0	0
$421$	9326.47	1.07968	0.539839	−	0.841768i	$-0.318485\pi$
$421$	9326.47	1.07968	0.539839	+	0.841768i	$0.318485\pi$
$422$	0	0
$423$	−2256.20	−0.259339
$424$	0	0
$425$	21606.7	2.46607
$426$	0	0
$427$	517.525	0.0586530
$428$	0	0
$429$	866.960	0.0975693
$430$	0	0
$431$	−15498.1	−1.73205	−0.866027	−	0.499998i	$-0.833334\pi$
$431$	−15498.1	−1.73205	−0.866027	+	0.499998i	$0.833334\pi$
$432$	0	0
$433$	3861.65	0.428589	0.214294	−	0.976769i	$-0.431255\pi$
$433$	3861.65	0.428589	0.214294	+	0.976769i	$0.431255\pi$
$434$	0	0
$435$	1028.07	0.113315
$436$	0	0
$437$	5163.16	0.565189
$438$	0	0
$439$	4826.85	0.524767	0.262384	−	0.964964i	$-0.415491\pi$
$439$	4826.85	0.524767	0.262384	+	0.964964i	$0.415491\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−4796.48	−0.514419	−0.257209	−	0.966356i	$-0.582803\pi$
$443$	−4796.48	−0.514419	−0.257209	+	0.966356i	$0.582803\pi$
$444$	0	0
$445$	10337.3	1.10120
$446$	0	0
$447$	−422.430	−0.0446985
$448$	0	0
$449$	−10161.7	−1.06806	−0.534029	−	0.845466i	$-0.679323\pi$
$449$	−10161.7	−1.06806	−0.534029	+	0.845466i	$0.679323\pi$
$450$	0	0
$451$	358.606	0.0374415
$452$	0	0
$453$	−7479.01	−0.775705
$454$	0	0
$455$	3842.91	0.395952
$456$	0	0
$457$	10037.3	1.02741	0.513704	−	0.857968i	$-0.328273\pi$
$457$	10037.3	1.02741	0.513704	+	0.857968i	$0.328273\pi$
$458$	0	0
$459$	−1818.31	−0.184905
$460$	0	0
$461$	−12646.4	−1.27766	−0.638831	−	0.769347i	$-0.720582\pi$
$461$	−12646.4	−1.27766	−0.638831	+	0.769347i	$0.720582\pi$
$462$	0	0
$463$	17077.8	1.71419	0.857097	−	0.515155i	$-0.172266\pi$
$463$	17077.8	1.71419	0.857097	+	0.515155i	$0.172266\pi$
$464$	0	0
$465$	17682.6	1.76346
$466$	0	0
$467$	899.992	0.0891792	0.0445896	−	0.999005i	$-0.485802\pi$
$467$	899.992	0.0891792	0.0445896	+	0.999005i	$0.485802\pi$
$468$	0	0
$469$	5918.12	0.582673
$470$	0	0
$471$	−8043.85	−0.786924
$472$	0	0
$473$	3132.60	0.304518
$474$	0	0
$475$	13548.9	1.30877
$476$	0	0
$477$	−250.542	−0.0240494
$478$	0	0
$479$	5953.85	0.567929	0.283965	−	0.958835i	$-0.408350\pi$
$479$	5953.85	0.567929	0.283965	+	0.958835i	$0.408350\pi$
$480$	0	0
$481$	−3219.79	−0.305218
$482$	0	0
$483$	−2567.54	−0.241878
$484$	0	0
$485$	25007.7	2.34132
$486$	0	0
$487$	−13309.9	−1.23846	−0.619229	−	0.785211i	$-0.712555\pi$
$487$	−13309.9	−1.23846	−0.619229	+	0.785211i	$0.712555\pi$
$488$	0	0
$489$	−10875.8	−1.00577
$490$	0	0
$491$	1894.37	0.174117	0.0870587	−	0.996203i	$-0.472253\pi$
$491$	1894.37	0.174117	0.0870587	+	0.996203i	$0.472253\pi$
$492$	0	0
$493$	−1092.99	−0.0998492
$494$	0	0
$495$	−2112.20	−0.191791
$496$	0	0
$497$	5174.75	0.467041
$498$	0	0
$499$	5338.14	0.478894	0.239447	−	0.970909i	$-0.423034\pi$
$499$	5338.14	0.478894	0.239447	+	0.970909i	$0.423034\pi$
$500$	0	0
$501$	3728.47	0.332487
$502$	0	0
$503$	−6895.58	−0.611250	−0.305625	−	0.952152i	$-0.598865\pi$
$503$	−6895.58	−0.611250	−0.305625	+	0.952152i	$0.598865\pi$
$504$	0	0
$505$	32824.4	2.89241
$506$	0	0
$507$	4563.00	0.399704
$508$	0	0
$509$	1638.74	0.142703	0.0713516	−	0.997451i	$-0.477269\pi$
$509$	1638.74	0.142703	0.0713516	+	0.997451i	$0.477269\pi$
$510$	0	0
$511$	−4915.42	−0.425529
$512$	0	0
$513$	−1140.20	−0.0981310
$514$	0	0
$515$	−37584.8	−3.21589
$516$	0	0
$517$	2786.38	0.237031
$518$	0	0
$519$	−749.333	−0.0633759
$520$	0	0
$521$	13423.8	1.12880	0.564401	−	0.825501i	$-0.309107\pi$
$521$	13423.8	1.12880	0.564401	+	0.825501i	$0.309107\pi$
$522$	0	0
$523$	−15758.4	−1.31753	−0.658763	−	0.752350i	$-0.728920\pi$
$523$	−15758.4	−1.31753	−0.658763	+	0.752350i	$0.728920\pi$
$524$	0	0
$525$	−6737.60	−0.560101
$526$	0	0
$527$	−18799.2	−1.55390
$528$	0	0
$529$	2781.40	0.228602
$530$	0	0
$531$	6101.15	0.498620
$532$	0	0
$533$	−838.855	−0.0681704
$534$	0	0
$535$	24078.8	1.94583
$536$	0	0
$537$	10588.4	0.850884
$538$	0	0
$539$	−544.629	−0.0435229
$540$	0	0
$541$	−12144.5	−0.965127	−0.482564	−	0.875861i	$-0.660294\pi$
$541$	−12144.5	−0.965127	−0.482564	+	0.875861i	$0.660294\pi$
$542$	0	0
$543$	7079.20	0.559480
$544$	0	0
$545$	44123.3	3.46795
$546$	0	0
$547$	11271.8	0.881070	0.440535	−	0.897735i	$-0.354789\pi$
$547$	11271.8	0.881070	0.440535	+	0.897735i	$0.354789\pi$
$548$	0	0
$549$	665.390	0.0517270
$550$	0	0
$551$	−685.379	−0.0529911
$552$	0	0
$553$	1754.82	0.134942
$554$	0	0
$555$	7844.47	0.599963
$556$	0	0
$557$	−13000.5	−0.988954	−0.494477	−	0.869191i	$-0.664640\pi$
$557$	−13000.5	−0.988954	−0.494477	+	0.869191i	$0.664640\pi$
$558$	0	0
$559$	−7327.79	−0.554441
$560$	0	0
$561$	2245.58	0.168999
$562$	0	0
$563$	−18246.0	−1.36586	−0.682929	−	0.730485i	$-0.739294\pi$
$563$	−18246.0	−1.36586	−0.682929	+	0.730485i	$0.739294\pi$
$564$	0	0
$565$	−7016.42	−0.522448
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	10249.3	0.755140	0.377570	−	0.925981i	$-0.376760\pi$
$569$	10249.3	0.755140	0.377570	+	0.925981i	$0.376760\pi$
$570$	0	0
$571$	−1383.57	−0.101402	−0.0507011	−	0.998714i	$-0.516146\pi$
$571$	−1383.57	−0.101402	−0.0507011	+	0.998714i	$0.516146\pi$
$572$	0	0
$573$	2060.58	0.150230
$574$	0	0
$575$	39226.8	2.84499
$576$	0	0
$577$	−346.308	−0.0249861	−0.0124931	−	0.999922i	$-0.503977\pi$
$577$	−346.308	−0.0249861	−0.0124931	+	0.999922i	$0.503977\pi$
$578$	0	0
$579$	4370.68	0.313712
$580$	0	0
$581$	−4895.46	−0.349566
$582$	0	0
$583$	309.416	0.0219806
$584$	0	0
$585$	4940.88	0.349197
$586$	0	0
$587$	−8208.65	−0.577184	−0.288592	−	0.957452i	$-0.593187\pi$
$587$	−8208.65	−0.577184	−0.288592	+	0.957452i	$0.593187\pi$
$588$	0	0
$589$	−11788.4	−0.824672
$590$	0	0
$591$	14225.1	0.990090
$592$	0	0
$593$	19522.9	1.35196	0.675978	−	0.736922i	$-0.263721\pi$
$593$	19522.9	1.35196	0.675978	+	0.736922i	$0.263721\pi$
$594$	0	0
$595$	9953.82	0.685826
$596$	0	0
$597$	−1689.81	−0.115845
$598$	0	0
$599$	294.580	0.0200938	0.0100469	−	0.999950i	$-0.496802\pi$
$599$	294.580	0.0200938	0.0100469	+	0.999950i	$0.496802\pi$
$600$	0	0
$601$	1132.24	0.0768473	0.0384236	−	0.999262i	$-0.487766\pi$
$601$	1132.24	0.0768473	0.0384236	+	0.999262i	$0.487766\pi$
$602$	0	0
$603$	7609.02	0.513869
$604$	0	0
$605$	−25495.4	−1.71328
$606$	0	0
$607$	2175.61	0.145478	0.0727390	−	0.997351i	$-0.476826\pi$
$607$	2175.61	0.145478	0.0727390	+	0.997351i	$0.476826\pi$
$608$	0	0
$609$	340.825	0.0226780
$610$	0	0
$611$	−6517.92	−0.431566
$612$	0	0
$613$	23243.3	1.53147	0.765734	−	0.643158i	$-0.222376\pi$
$613$	23243.3	1.53147	0.765734	+	0.643158i	$0.222376\pi$
$614$	0	0
$615$	2043.73	0.134002
$616$	0	0
$617$	−23917.8	−1.56060	−0.780302	−	0.625403i	$-0.784935\pi$
$617$	−23917.8	−1.56060	−0.780302	+	0.625403i	$0.784935\pi$
$618$	0	0
$619$	14915.1	0.968482	0.484241	−	0.874935i	$-0.339096\pi$
$619$	14915.1	0.968482	0.484241	+	0.874935i	$0.339096\pi$
$620$	0	0
$621$	−3301.12	−0.213316
$622$	0	0
$623$	3427.02	0.220386
$624$	0	0
$625$	47207.3	3.02127
$626$	0	0
$627$	1408.14	0.0896898
$628$	0	0
$629$	−8339.83	−0.528666
$630$	0	0
$631$	−14782.3	−0.932604	−0.466302	−	0.884626i	$-0.654414\pi$
$631$	−14782.3	−0.932604	−0.466302	+	0.884626i	$0.654414\pi$
$632$	0	0
$633$	−9723.57	−0.610549
$634$	0	0
$635$	−26641.0	−1.66491
$636$	0	0
$637$	1274.00	0.0792429
$638$	0	0
$639$	6653.25	0.411891
$640$	0	0
$641$	3820.66	0.235424	0.117712	−	0.993048i	$-0.462444\pi$
$641$	3820.66	0.235424	0.117712	+	0.993048i	$0.462444\pi$
$642$	0	0
$643$	31644.8	1.94082	0.970411	−	0.241460i	$-0.0776264\pi$
$643$	31644.8	1.94082	0.970411	+	0.241460i	$0.0776264\pi$
$644$	0	0
$645$	17852.9	1.08986
$646$	0	0
$647$	28264.2	1.71744	0.858718	−	0.512448i	$-0.171261\pi$
$647$	28264.2	1.71744	0.858718	+	0.512448i	$0.171261\pi$
$648$	0	0
$649$	−7534.84	−0.455729
$650$	0	0
$651$	5862.12	0.352926
$652$	0	0
$653$	−10647.4	−0.638076	−0.319038	−	0.947742i	$-0.603360\pi$
$653$	−10647.4	−0.638076	−0.319038	+	0.947742i	$0.603360\pi$
$654$	0	0
$655$	−55722.8	−3.32407
$656$	0	0
$657$	−6319.83	−0.375282
$658$	0	0
$659$	−22956.8	−1.35701	−0.678505	−	0.734595i	$-0.737372\pi$
$659$	−22956.8	−1.35701	−0.678505	+	0.734595i	$0.737372\pi$
$660$	0	0
$661$	11133.8	0.655149	0.327575	−	0.944825i	$-0.393769\pi$
$661$	11133.8	0.655149	0.327575	+	0.944825i	$0.393769\pi$
$662$	0	0
$663$	−5252.88	−0.307700
$664$	0	0
$665$	6241.73	0.363976
$666$	0	0
$667$	−1984.31	−0.115192
$668$	0	0
$669$	−7969.94	−0.460592
$670$	0	0
$671$	−821.747	−0.0472775
$672$	0	0
$673$	5327.32	0.305131	0.152565	−	0.988293i	$-0.451247\pi$
$673$	5327.32	0.305131	0.152565	+	0.988293i	$0.451247\pi$
$674$	0	0
$675$	−8662.63	−0.493962
$676$	0	0
$677$	15023.5	0.852878	0.426439	−	0.904516i	$-0.359768\pi$
$677$	15023.5	0.852878	0.426439	+	0.904516i	$0.359768\pi$
$678$	0	0
$679$	8290.56	0.468575
$680$	0	0
$681$	−9375.05	−0.527537
$682$	0	0
$683$	14635.9	0.819954	0.409977	−	0.912096i	$-0.365537\pi$
$683$	14635.9	0.819954	0.409977	+	0.912096i	$0.365537\pi$
$684$	0	0
$685$	−42233.5	−2.35571
$686$	0	0
$687$	1209.32	0.0671594
$688$	0	0
$689$	−723.789	−0.0400206
$690$	0	0
$691$	7115.23	0.391717	0.195858	−	0.980632i	$-0.437251\pi$
$691$	7115.23	0.391717	0.195858	+	0.980632i	$0.437251\pi$
$692$	0	0
$693$	−700.237	−0.0383836
$694$	0	0
$695$	42544.2	2.32201
$696$	0	0
$697$	−2172.78	−0.118078
$698$	0	0
$699$	3469.41	0.187733
$700$	0	0
$701$	14927.8	0.804299	0.402149	−	0.915574i	$-0.368263\pi$
$701$	14927.8	0.804299	0.402149	+	0.915574i	$0.368263\pi$
$702$	0	0
$703$	−5229.65	−0.280569
$704$	0	0
$705$	15879.8	0.848324
$706$	0	0
$707$	10881.9	0.578864
$708$	0	0
$709$	888.426	0.0470600	0.0235300	−	0.999723i	$-0.492509\pi$
$709$	888.426	0.0470600	0.0235300	+	0.999723i	$0.492509\pi$
$710$	0	0
$711$	2256.20	0.119007
$712$	0	0
$713$	−34129.7	−1.79266
$714$	0	0
$715$	−6101.92	−0.319159
$716$	0	0
$717$	−11199.1	−0.583315
$718$	0	0
$719$	−24155.9	−1.25294	−0.626470	−	0.779445i	$-0.715501\pi$
$719$	−24155.9	−1.25294	−0.626470	+	0.779445i	$0.715501\pi$
$720$	0	0
$721$	−12460.1	−0.643604
$722$	0	0
$723$	−9459.13	−0.486568
$724$	0	0
$725$	−5207.12	−0.266742
$726$	0	0
$727$	11284.0	0.575654	0.287827	−	0.957682i	$-0.407067\pi$
$727$	11284.0	0.575654	0.287827	+	0.957682i	$0.407067\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−18980.3	−0.960343
$732$	0	0
$733$	19307.2	0.972887	0.486444	−	0.873712i	$-0.338294\pi$
$733$	19307.2	0.972887	0.486444	+	0.873712i	$0.338294\pi$
$734$	0	0
$735$	−3103.89	−0.155767
$736$	0	0
$737$	−9397.03	−0.469666
$738$	0	0
$739$	−16139.9	−0.803406	−0.401703	−	0.915770i	$-0.631582\pi$
$739$	−16139.9	−0.803406	−0.401703	+	0.915770i	$0.631582\pi$
$740$	0	0
$741$	−3293.92	−0.163300
$742$	0	0
$743$	−6613.88	−0.326568	−0.163284	−	0.986579i	$-0.552209\pi$
$743$	−6613.88	−0.326568	−0.163284	+	0.986579i	$0.552209\pi$
$744$	0	0
$745$	2973.18	0.146213
$746$	0	0
$747$	−6294.17	−0.308289
$748$	0	0
$749$	7982.61	0.389424
$750$	0	0
$751$	−5167.07	−0.251064	−0.125532	−	0.992090i	$-0.540064\pi$
$751$	−5167.07	−0.251064	−0.125532	+	0.992090i	$0.540064\pi$
$752$	0	0
$753$	7324.34	0.354467
$754$	0	0
$755$	52639.4	2.53741
$756$	0	0
$757$	−21745.4	−1.04405	−0.522027	−	0.852929i	$-0.674824\pi$
$757$	−21745.4	−1.04405	−0.522027	+	0.852929i	$0.674824\pi$
$758$	0	0
$759$	4076.84	0.194967
$760$	0	0
$761$	3002.22	0.143010	0.0715049	−	0.997440i	$-0.477220\pi$
$761$	3002.22	0.143010	0.0715049	+	0.997440i	$0.477220\pi$
$762$	0	0
$763$	14627.7	0.694049
$764$	0	0
$765$	12797.8	0.604842
$766$	0	0
$767$	17625.6	0.829755
$768$	0	0
$769$	−4008.18	−0.187957	−0.0939784	−	0.995574i	$-0.529958\pi$
$769$	−4008.18	−0.187957	−0.0939784	+	0.995574i	$0.529958\pi$
$770$	0	0
$771$	−7249.65	−0.338638
$772$	0	0
$773$	3816.48	0.177580	0.0887900	−	0.996050i	$-0.471700\pi$
$773$	3816.48	0.177580	0.0887900	+	0.996050i	$0.471700\pi$
$774$	0	0
$775$	−89561.5	−4.15116
$776$	0	0
$777$	2600.60	0.120072
$778$	0	0
$779$	−1362.49	−0.0626651
$780$	0	0
$781$	−8216.68	−0.376461
$782$	0	0
$783$	438.203	0.0200001
$784$	0	0
$785$	56615.0	2.57411
$786$	0	0
$787$	−22492.0	−1.01874	−0.509372	−	0.860546i	$-0.670122\pi$
$787$	−22492.0	−1.01874	−0.509372	+	0.860546i	$0.670122\pi$
$788$	0	0
$789$	−1528.28	−0.0689585
$790$	0	0
$791$	−2326.08	−0.104559
$792$	0	0
$793$	1922.24	0.0860790
$794$	0	0
$795$	1763.39	0.0786679
$796$	0	0
$797$	41001.1	1.82225	0.911124	−	0.412132i	$-0.135216\pi$
$797$	41001.1	1.82225	0.911124	+	0.412132i	$0.135216\pi$
$798$	0	0
$799$	−16882.6	−0.747513
$800$	0	0
$801$	4406.17	0.194362
$802$	0	0
$803$	7804.90	0.343000
$804$	0	0
$805$	18071.1	0.791207
$806$	0	0
$807$	24928.3	1.08738
$808$	0	0
$809$	−30507.0	−1.32579	−0.662897	−	0.748710i	$-0.730673\pi$
$809$	−30507.0	−1.32579	−0.662897	+	0.748710i	$0.730673\pi$
$810$	0	0
$811$	−3799.66	−0.164518	−0.0822590	−	0.996611i	$-0.526213\pi$
$811$	−3799.66	−0.164518	−0.0822590	+	0.996611i	$0.526213\pi$
$812$	0	0
$813$	4499.64	0.194107
$814$	0	0
$815$	76546.9	3.28996
$816$	0	0
$817$	−11902.0	−0.509665
$818$	0	0
$819$	1638.00	0.0698857
$820$	0	0
$821$	27444.1	1.16663	0.583316	−	0.812245i	$-0.301755\pi$
$821$	27444.1	1.16663	0.583316	+	0.812245i	$0.301755\pi$
$822$	0	0
$823$	9422.95	0.399105	0.199553	−	0.979887i	$-0.436051\pi$
$823$	9422.95	0.399105	0.199553	+	0.979887i	$0.436051\pi$
$824$	0	0
$825$	10698.2	0.451472
$826$	0	0
$827$	−5531.49	−0.232586	−0.116293	−	0.993215i	$-0.537101\pi$
$827$	−5531.49	−0.232586	−0.116293	+	0.993215i	$0.537101\pi$
$828$	0	0
$829$	47025.9	1.97018	0.985088	−	0.172050i	$-0.0550391\pi$
$829$	47025.9	1.97018	0.985088	+	0.172050i	$0.0550391\pi$
$830$	0	0
$831$	−5172.08	−0.215905
$832$	0	0
$833$	3299.89	0.137256
$834$	0	0
$835$	−26242.1	−1.08760
$836$	0	0
$837$	7537.02	0.311251
$838$	0	0
$839$	25249.1	1.03897	0.519484	−	0.854480i	$-0.326124\pi$
$839$	25249.1	1.03897	0.519484	+	0.854480i	$0.326124\pi$
$840$	0	0
$841$	−24125.6	−0.989200
$842$	0	0
$843$	25179.7	1.02875
$844$	0	0
$845$	−32115.7	−1.30747
$846$	0	0
$847$	−8452.22	−0.342883
$848$	0	0
$849$	−24150.0	−0.976236
$850$	0	0
$851$	−15140.9	−0.609898
$852$	0	0
$853$	−5220.44	−0.209548	−0.104774	−	0.994496i	$-0.533412\pi$
$853$	−5220.44	−0.209548	−0.104774	+	0.994496i	$0.533412\pi$
$854$	0	0
$855$	8025.08	0.320997
$856$	0	0
$857$	−16752.7	−0.667751	−0.333876	−	0.942617i	$-0.608357\pi$
$857$	−16752.7	−0.667751	−0.333876	+	0.942617i	$0.608357\pi$
$858$	0	0
$859$	−46840.6	−1.86051	−0.930257	−	0.366908i	$-0.880416\pi$
$859$	−46840.6	−1.86051	−0.930257	+	0.366908i	$0.880416\pi$
$860$	0	0
$861$	677.537	0.0268181
$862$	0	0
$863$	2861.77	0.112881	0.0564403	−	0.998406i	$-0.482025\pi$
$863$	2861.77	0.112881	0.0564403	+	0.998406i	$0.482025\pi$
$864$	0	0
$865$	5274.03	0.207309
$866$	0	0
$867$	1133.10	0.0443854
$868$	0	0
$869$	−2786.38	−0.108770
$870$	0	0
$871$	21981.6	0.855130
$872$	0	0
$873$	10659.3	0.413244
$874$	0	0
$875$	28945.7	1.11833
$876$	0	0
$877$	−28321.1	−1.09046	−0.545231	−	0.838286i	$-0.683558\pi$
$877$	−28321.1	−1.09046	−0.545231	+	0.838286i	$0.683558\pi$
$878$	0	0
$879$	−29660.0	−1.13812
$880$	0	0
$881$	−17763.2	−0.679293	−0.339647	−	0.940553i	$-0.610307\pi$
$881$	−17763.2	−0.679293	−0.339647	+	0.940553i	$0.610307\pi$
$882$	0	0
$883$	5366.85	0.204540	0.102270	−	0.994757i	$-0.467389\pi$
$883$	5366.85	0.204540	0.102270	+	0.994757i	$0.467389\pi$
$884$	0	0
$885$	−42941.7	−1.63104
$886$	0	0
$887$	−50485.4	−1.91109	−0.955544	−	0.294849i	$-0.904731\pi$
$887$	−50485.4	−1.91109	−0.955544	+	0.294849i	$0.904731\pi$
$888$	0	0
$889$	−8832.02	−0.333202
$890$	0	0
$891$	−900.305	−0.0338511
$892$	0	0
$893$	−10586.5	−0.396714
$894$	0	0
$895$	−74524.5	−2.78333
$896$	0	0
$897$	−9536.56	−0.354980
$898$	0	0
$899$	4530.52	0.168077
$900$	0	0
$901$	−1874.74	−0.0693193
$902$	0	0
$903$	5918.60	0.218116
$904$	0	0
$905$	−49825.5	−1.83012
$906$	0	0
$907$	14517.9	0.531487	0.265744	−	0.964044i	$-0.414382\pi$
$907$	14517.9	0.531487	0.265744	+	0.964044i	$0.414382\pi$
$908$	0	0
$909$	13991.1	0.510510
$910$	0	0
$911$	39837.6	1.44883	0.724413	−	0.689367i	$-0.242111\pi$
$911$	39837.6	1.44883	0.724413	+	0.689367i	$0.242111\pi$
$912$	0	0
$913$	7773.21	0.281770
$914$	0	0
$915$	−4683.21	−0.169204
$916$	0	0
$917$	−18473.2	−0.665255
$918$	0	0
$919$	3406.72	0.122282	0.0611411	−	0.998129i	$-0.480526\pi$
$919$	3406.72	0.122282	0.0611411	+	0.998129i	$0.480526\pi$
$920$	0	0
$921$	−5163.44	−0.184735
$922$	0	0
$923$	19220.5	0.685429
$924$	0	0
$925$	−39732.0	−1.41230
$926$	0	0
$927$	−16020.1	−0.567605
$928$	0	0
$929$	−7453.31	−0.263224	−0.131612	−	0.991301i	$-0.542015\pi$
$929$	−7453.31	−0.263224	−0.131612	+	0.991301i	$0.542015\pi$
$930$	0	0
$931$	2069.26	0.0728434
$932$	0	0
$933$	13253.8	0.465070
$934$	0	0
$935$	−15805.1	−0.552814
$936$	0	0
$937$	−13599.3	−0.474140	−0.237070	−	0.971493i	$-0.576187\pi$
$937$	−13599.3	−0.474140	−0.237070	+	0.971493i	$0.576187\pi$
$938$	0	0
$939$	25053.3	0.870694
$940$	0	0
$941$	−1430.50	−0.0495569	−0.0247784	−	0.999693i	$-0.507888\pi$
$941$	−1430.50	−0.0495569	−0.0247784	+	0.999693i	$0.507888\pi$
$942$	0	0
$943$	−3944.67	−0.136221
$944$	0	0
$945$	−3990.71	−0.137373
$946$	0	0
$947$	−7382.05	−0.253310	−0.126655	−	0.991947i	$-0.540424\pi$
$947$	−7382.05	−0.253310	−0.126655	+	0.991947i	$0.540424\pi$
$948$	0	0
$949$	−18257.3	−0.624506
$950$	0	0
$951$	−15007.6	−0.511728
$952$	0	0
$953$	19941.7	0.677833	0.338916	−	0.940816i	$-0.389940\pi$
$953$	19941.7	0.677833	0.338916	+	0.940816i	$0.389940\pi$
$954$	0	0
$955$	−14502.9	−0.491418
$956$	0	0
$957$	−541.175	−0.0182797
$958$	0	0
$959$	−14001.2	−0.471453
$960$	0	0
$961$	48133.0	1.61569
$962$	0	0
$963$	10263.4	0.343439
$964$	0	0
$965$	−30762.1	−1.02618
$966$	0	0
$967$	−6161.51	−0.204903	−0.102451	−	0.994738i	$-0.532669\pi$
$967$	−6161.51	−0.204903	−0.102451	+	0.994738i	$0.532669\pi$
$968$	0	0
$969$	−8531.84	−0.282851
$970$	0	0
$971$	−35313.3	−1.16710	−0.583552	−	0.812076i	$-0.698338\pi$
$971$	−35313.3	−1.16710	−0.583552	+	0.812076i	$0.698338\pi$
$972$	0	0
$973$	14104.2	0.464709
$974$	0	0
$975$	−25025.4	−0.822003
$976$	0	0
$977$	9684.81	0.317139	0.158569	−	0.987348i	$-0.449312\pi$
$977$	9684.81	0.317139	0.158569	+	0.987348i	$0.449312\pi$
$978$	0	0
$979$	−5441.56	−0.177643
$980$	0	0
$981$	18807.1	0.612094
$982$	0	0
$983$	52986.8	1.71924	0.859621	−	0.510932i	$-0.170699\pi$
$983$	52986.8	1.71924	0.859621	+	0.510932i	$0.170699\pi$
$984$	0	0
$985$	−100121.	−3.23869
$986$	0	0
$987$	5264.47	0.169777
$988$	0	0
$989$	−34458.5	−1.10790
$990$	0	0
$991$	6415.80	0.205656	0.102828	−	0.994699i	$-0.467211\pi$
$991$	6415.80	0.205656	0.102828	+	0.994699i	$0.467211\pi$
$992$	0	0
$993$	3632.77	0.116095
$994$	0	0
$995$	11893.4	0.378940
$996$	0	0
$997$	−20825.7	−0.661540	−0.330770	−	0.943711i	$-0.607308\pi$
$997$	−20825.7	−0.661540	−0.330770	+	0.943711i	$0.607308\pi$
$998$	0	0
$999$	3343.63	0.105894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.a.i.1.2	✓	2
3.2	odd	2		2016.4.a.h.1.1		2
4.3	odd	2		672.4.a.n.1.2	yes	2
8.3	odd	2		1344.4.a.bc.1.1		2
8.5	even	2		1344.4.a.bk.1.1		2
12.11	even	2		2016.4.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.i.1.2	✓	2	1.1	even	1	trivial
672.4.a.n.1.2	yes	2	4.3	odd	2
1344.4.a.bc.1.1		2	8.3	odd	2
1344.4.a.bk.1.1		2	8.5	even	2
2016.4.a.g.1.1		2	12.11	even	2
2016.4.a.h.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +21.1149 q^{5} +7.00000 q^{7} +9.00000 q^{9} -11.1149 q^{11} +26.0000 q^{13} -63.3446 q^{15} +67.3446 q^{17} +42.2298 q^{19} -21.0000 q^{21} +122.264 q^{23} +320.838 q^{25} -27.0000 q^{27} -16.2298 q^{29} -279.149 q^{31} +33.3446 q^{33} +147.804 q^{35} -123.838 q^{37} -78.0000 q^{39} -32.2636 q^{41} -281.838 q^{43} +190.034 q^{45} -250.689 q^{47} +49.0000 q^{49} -202.034 q^{51} -27.8380 q^{53} -234.689 q^{55} -126.689 q^{57} +677.906 q^{59} +73.9322 q^{61} +63.0000 q^{63} +548.987 q^{65} +845.446 q^{67} -366.791 q^{69} +739.250 q^{71} -702.203 q^{73} -962.514 q^{75} -77.8041 q^{77} +250.689 q^{79} +81.0000 q^{81} -699.352 q^{83} +1421.97 q^{85} +48.6893 q^{87} +489.574 q^{89} +182.000 q^{91} +837.446 q^{93} +891.676 q^{95} +1184.37 q^{97} -100.034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 16 q^{5} + 14 q^{7} + 18 q^{9} + 4 q^{11} + 52 q^{13} - 48 q^{15} + 56 q^{17} + 32 q^{19} - 42 q^{21} - 44 q^{23} + 222 q^{25} - 54 q^{27} + 20 q^{29} - 296 q^{31} - 12 q^{33} + 112 q^{35}+ \cdots + 36 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 16 * q^5 + 14 * q^7 + 18 * q^9 + 4 * q^11 + 52 * q^13 - 48 * q^15 + 56 * q^17 + 32 * q^19 - 42 * q^21 - 44 * q^23 + 222 * q^25 - 54 * q^27 + 20 * q^29 - 296 * q^31 - 12 * q^33 + 112 * q^35 + 172 * q^37 - 156 * q^39 + 224 * q^41 - 144 * q^43 + 144 * q^45 - 344 * q^47 + 98 * q^49 - 168 * q^51 + 364 * q^53 - 312 * q^55 - 96 * q^57 + 464 * q^59 + 620 * q^61 + 126 * q^63 + 416 * q^65 + 904 * q^67 + 132 * q^69 + 508 * q^71 + 12 * q^73 - 666 * q^75 + 28 * q^77 + 344 * q^79 + 162 * q^81 + 280 * q^83 + 1480 * q^85 - 60 * q^87 + 848 * q^89 + 364 * q^91 + 888 * q^93 + 944 * q^95 + 1372 * q^97 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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